PHYSICAL REVIEW A 95, 043826 (2017)

Emergent equilibrium in many-body optical bistability

M. Foss-Feig,1,2,3 P. Niroula,2,4 J. T. Young,2 M. Hafezi,2,5 A. V. Gorshkov,2,3 R. M. Wilson,6 and M. F. Maghrebi2,3,7

1United States Army Research Laboratory, Adelphi, Maryland 20783, USA2Joint Quantum Institute, NIST and University of Maryland, College Park, Maryland 20742, USA

3Joint Center for Quantum Information and Computer Science, NIST and University of Maryland, College Park, Maryland 20742, USA4Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

5Department of Electrical and Computer Engineering and Institute for Research in Electronics and Applied Physics,University of Maryland, College Park, Maryland 20742, USA

6Department of Physics, United States Naval Academy, Annapolis, Maryland 21402, USA7Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA

(Received 3 November 2016; revised manuscript received 10 January 2017; published 17 April 2017)

Many-body systems constructed of quantum-optical building blocks can now be realized in experimentalplatforms ranging from exciton-polariton fluids to ultracold Rydberg gases, establishing a fascinating interfacebetween traditional many-body physics and the driven-dissipative, nonequilibrium setting of cavity QED. Atthis interface, the standard techniques and intuitions of both fields are called into question, obscuring issuesas fundamental as the role of fluctuations, dimensionality, and symmetry on the nature of collective behaviorand phase transitions. Here, we study the driven-dissipative Bose-Hubbard model, a minimal description ofnumerous atomic, optical, and solid-state systems in which particle loss is countered by coherent driving.Despite being a lattice version of optical bistability, a foundational and patently nonequilibrium model ofcavity QED, the steady state possesses an emergent equilibrium description in terms of a classical Ising model.We establish this picture by making new connections between traditional techniques from many-body physics(functional integrals) and quantum optics (the system-size expansion). To lowest order in a controlled expansion—organized around the experimentally relevant limit of weak interactions—the full quantum dynamics reduces tononequilibrium Langevin equations, which support a phase transition described by model A of the Hohenberg-Halperin classification. Numerical simulations of the Langevin equations corroborate this picture, revealing thatcanonical behavior associated with the Ising model manifests readily in simple experimental observables.

DOI: 10.1103/PhysRevA.95.043826

I. INTRODUCTION

While systems described by cavity quantum electrody-namics (QED) often contain many interacting degrees offreedom, they are unconventional from the standpoint oftraditional many-body physics for two primary reasons. First,the mediation of interactions through a small number ofdelocalized cavity modes generally leads to extremely long-ranged interactions [1], which suppress the role of fluctuationsand often enable accurate mean-field descriptions. In this sensethey are simpler than conventional solid-state realizationsof many-body physics, in which short-range interactionspromote both quantum and thermal fluctuations to an importantrole, especially in low spatial dimensions [2,3]. Second,cavity-QED systems are typically driven and dissipative; asa result, even if they reach a time-independent steady statethey will generally not be in thermal equilibrium [4]. In thissense they are more complicated than conventional solid-staterealizations of many-body physics, in which coupling to athermal reservoir is typically assumed and well justified,leaving the system in thermal equilibrium and enabling thepowerful tools of statistical mechanics to be employed [5].

In recent years, experimental advances in quantum opticshave begun to blur the first of these distinctions [6–9], withplatforms including exciton-polariton fluids in semiconductorquantum wells [10–15], circuit QED [16–20], optical fibers,waveguides, and photonic crystals [21–26], small-mode-volume optical resonators [27,28], and Rydberg ensembles[29–32] all making progress towards realizing large-scale ar-rays of short-range coupled quantum-optical building blocks.

These developments have led many researchers to revisitfundamental questions surrounding the fate of nonequilibriumquantum-optical systems in situations where, due to theimportance of either dissipative or quantum fluctuations, amean-field description is insufficient [7,15,33–44]. The pri-mary goal of this paper is to elucidate the physics of a canonicalmany-body model made relevant by these developments—the driven-dissipative Bose-Hubbard model [45–48]—whichfurnishes a minimal description of, e.g., coherently drivenexciton-polariton fluids confined in coupled microcavities orother patterned semiconductor devices [14,15,49–51]. In thatcontext, the weak-coupling limit of the model is by far the mostexperimentally relevant. However, as will be discussed, neitherperturbation theory nor mean-field theory are sufficient forcapturing the weak-coupling physics, even qualitatively, norcan it be inferred from the well-studied equilibrium physics ofthe Bose-Hubbard model.

Nevertheless, by exploiting connections between ideasfrom many-body physics (functional-integral treatments ofnonequilibrium field theory) and quantum optics (phase-spacetechniques and the system-size expansion), we identify aquantitatively accurate mapping of the weak-coupling limit ofthe driven-dissipative Bose-Hubbard model onto a field theorygoverning the equilibrium physics of the finite-temperatureclassical Ising model. In this way (cf. our main results inSec. V), we are able to make quantitative predictions forthe steady-state phases and phase transitions of the modelin a parameter regime that is highly relevant to ongoingexperiments with exciton-polariton fluids [14,15].

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M. FOSS-FEIG et al. PHYSICAL REVIEW A 95, 043826 (2017)

Much of the previous work on the driven-dissipative Bose-Hubbard model has grown out of early proposals to simulatethe equilibrium Bose-Hubbard model in photonic systems,either in the transient regime of very weakly dissipative sys-tems [45,52], or through clever strategies to mitigate the effectsof particle loss [53–55]. In this context, the driven-dissipativemodel has been considered in an attempt to understand thecorruption of equilibrium physics by nonvanishing dissipationin realistic systems, and to identify qualitative signatures ofequilibrium Bose-Hubbard physics—e.g., fermionization forstrong interactions [46,56] or the incompressibility of thezero-temperature Mott-insulating phase [57–59]—that survivein steady state. The general spirit of this approach is tostart from the intuitions and expectations appropriate forthe equilibrium Bose-Hubbard model, and to build outwardtoward an understanding of the driven-dissipative dynamics;numerous interesting connections to the equilibrium physicsof the Bose-Hubbard model [34,46,60], as well as a varietyof surprising and genuinely nonequilibrium effects [17,46,61–64], have been discovered in this manner. But there are manyreasons to expect that the search for universal features ofthe driven-dissipative model benefits from, and perhaps evenrequires, a fundamentally different approach. For example,the ground-state and thermal phase transitions of the Bose-Hubbard model are intimately related to U(1) symmetry andthe associated particle-number conservation [65]. While theformer can be preserved in a driven-dissipative context bypumping the cavities incoherently [6], the latter remainsabsent [66], calling into question whether—in the senseof phase transitions and universality—any properties of theBose-Hubbard model can survive the presence of driving anddissipation [42].

Here, we instead pursue an understanding of the driven-dissipative Bose-Hubbard model from the ground up, start-ing from the well-understood nonequilibrium physics of asingle cavity, namely optical bistability [67], and adoptinga functional-integral formalism that is well suited to extendingthe essential single-cavity physics to a many-body setting[7,34,39,68]. As we show, the breaking of conservation lawsand symmetries at the microscopic level leads to universalproperties of the steady state that bear essentially no re-semblance to the equilibrium physics of the Bose-Hubbardmodel [57], but neither do they retain the fundamentallynonequilibrium character of optical bistability. Instead, thesteady state of the driven-dissipative Bose-Hubbard modeladmits an emergent equilibrium description in terms of afinite-temperature classical Ising model [69]. Specifically (seeFig. 1), two collective mean-field steady states are inheritedfrom the optical bistability of the individual cavities; they playthe role of the two local minima in the Ising model’s mean-field free energy, while dissipation (i.e., vacuum fluctuations)plays the role of thermal fluctuations, setting the effectivetemperature. By explicitly and quantitatively connecting twocanonical and minimal models of many-body physics—onea cornerstone of nonequilibrium quantum optics and one acornerstone of traditional equilibrium many-body physics—this paper provides a particularly simple and concrete exampleof the way in which equilibrium can emerge very naturallyfrom an a priori nonequilibrium many-body problem, evenwhen (a) mean-field theory fails and (b) the model with respect

FIG. 1. Summary of the correspondence between many-bodyoptical bistability and the classical Ising model that serves as itseffective equilibrium description. The two possible magnetizationsof the Ising model correspond to the bright and dark mean-fieldsteady states of the optically bistable cavities. All parameters andvariables are defined in the manuscript. In the bottom row, N isa parameter controlling the density scale, i.e., |ψ |2 ∼ N . Hencethe low-temperature limit of the Ising model corresponds to asemiclassical (large density) limit of optical bistability.

to which equilibrium emerges is not simply connected to theequilibrium physics of the underlying Hamiltonian.

We note that our conclusions are enabled by a unification ofideas from nonequilibrium field theory (originally applied to asimilar model in Ref. [69]) and quantum optics. In particular,we identify a small parameter 1/N related to the inverse“system size” (in the sense of the system-size expansion oftenemployed in quantum-optics studies of systems with only afew degrees of freedom), which controls the overall scale offluctuations, and thus the effective temperature. In the limit ofweak fluctuations, the qualitative predictions of Ref. [69] canbe justified, made quantitative, and even verified numerically.In this way, we not only identify the model with respectto which an effective thermal description emerges, but alsosemiquantitatively obtain the phase boundaries, effective tem-perature, and near-critical dynamics in terms of microscopicparameters. Crucially, the limit in which these methods areaccurate—weak coupling and large density—coincides withthe limit of the model that is most accessible experimentally.Thus, in addition to establishing a universal perspective on

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the physics of the driven-dissipative Bose-Hubbard model, thetechniques established here can be used to make quantitativepredictions for ongoing experiments with exciton-polaritonfluids.

Before proceeding, we caution that the emergence of aneffective equilibrium description as detailed in this paper,while potentially reasonably generic, should not be taken forgranted; other more genuinely nonequilibrium situations canand do arise in other models [35,39,41,70–72]. Ultimately, thegoal of this paper is not only to provide a detailed view into themechanisms by which thermal equilibrium can emerge fromthe microscopically nonequilibrium setting of many-bodycavity QED, nor by any means to insist that one must emerge,but also to establish and clarify deep connections betweenmany-body physics and quantum optics that may elucidatemore unusual behaviors made possible by the strong-couplingregime of quantum optics. It is natural to expect that analyses ofprototypical situations in which equilibrium does emerge willplay an important role in anticipating more exotic situations inwhich it does not.

After presenting the model and reviewing the well-knownsolution of the single-cavity case in Sec. II, our general strategyfor the many-body problem is laid out in Sec. III. Our approachis based on a well-established exact mapping of the masterequation onto a functional integral. Although it imposes someadditional notational burden, the functional-integral formalismhas the virtue of (1) being a convenient starting point for theidentification of approximation schemes, including controlledstrategies for going beyond mean-field theory [68], and (2) en-abling powerful techniques such as the renormalization groupto be applied [7]. Here, we use the functional integral to quicklyidentify an exactly solvable limit of the problem, around whicha semiclassical expansion (related to the system-size expansionof quantum optics) can be made. To leading nontrivial order inthis expansion, we obtain a quantitatively accurate mappingof the many-body quantum master equation onto classicalnonequilibrium Langevin equations, with a small parametercontrolling the strength of the noise. In Sec. IV we analyzethe mean-field equations of motion near the mean-field criticalpoint, which possess an emergent Z2 symmetry in the spiritof Ref. [73]. We show how the complex order parameterdecomposes into two real components, one of which staysmassive at the critical point and one of which does not. Byadiabatically eliminating the massive component, we arriveat a time-dependent Landau-Ginsburg equation for a scalarfield, which supports two different homogeneous solutionswithin the bistable region. Near the critical point and insidethe bistable region, we are able to analytically obtain the profileand velocity of domain walls separating domains of thesetwo different phases, and the vanishing of the domain-wallvelocity gives a zeroth-order approximation to the location ofa true (first-order) phase transition in more than one spatialdimension. In Sec. V we consider the effects of fluctuations inboth one and two spatial dimensions by (a) arguing that—nearthe critical point and for weak noise—the nonlinear Langevinequations become equivalent to model A of the Hohenberg-Halperin classification, and (b) solving the nonequilibriumLangevin equations numerically, which is valid even awayfrom the critical point. As our earlier analysis would suggest,the numerical results are qualitatively consistent with the

expected equilibrium physics of a classical Ising model ina longitudinal field. In one dimension, domains are seededby fluctuations, and the dynamics of their unbound domainwalls smooths the mean-field transition into a crossover. Intwo dimensions the domain walls exhibit a surface tension,enabling a line of true first-order phase transitions terminatingat a critical point.

II. MODEL

The model we consider can arise in a variety of contexts,but for concreteness we consider either a one-dimensional(1D) chain or a two-dimensional (2D) rectangular arrayof semiconductor microcavities supporting exciton-polaritons(see, e.g., Ref. [14]). We assume that the on-site energies ofexciton-polaritons are spatially uniform and equal to ω0, andthat the cavities are driven coherently and in phase by a laserwith frequency ωL. Upon making a unitary transformation toremove the time dependence of the driving, we obtain theHamiltonian

H = −J∑

⟨j,k⟩a†j ak − δ

∑

j

a†j aj + U

2

∑

j

a†j a

†j aj aj

+$∑

j

(aj + a†j ). (1)

Here, a†j (aj ) creates (annihilates) an exciton-polariton in the

j th cavity, J parametrizes the strength of a coherent couplingof exciton-polaritons between neighboring cavities, δ = ωL −ω0 is the detuning of the laser from cavity resonance, U sets thetwo-body interaction energy for exciton-polaritons confined inthe same cavity, and $ is the amplitude of the coherent driving.The notation ⟨j,k⟩ implies that the sum should be taken over allnearest-neighbor pairs of sites j and k. The driving is necessaryto stabilize a nontrivial steady state in the presence of particleloss out of the cavities at a rate κ . If the loss of exciton-polaritons is treated in the Born-Markov approximation, thedynamics of the combined unitary evolution under H and lossis described by a Markovian master equation [4],

dρ

dt= −i[H ,ρ] + κ

2

∑

j

(2aj ρa†j − ρa

†j aj − a

†j aj ρ). (2)

More generally, Eqs. (1) and (2) provide a natural (thoughcertainly not unique) generalization of the Bose-Hubbardmodel to the driven ($) and dissipative (κ) setting of quantumoptics.

The case of a single cavity has been thoroughly studiedin the quantum-optics literature, where it serves as a minimalmodel for dispersive optical bistability [67,74]. A mean-fielddescription of the problem can be obtained by writing down theequation of motion for ψ ≡ ⟨a⟩ and assuming that expectationvalues of normal-ordered operator products factorize (i.e.,making the replacement ⟨a†aa⟩ → |ψ |2ψ), giving

iψ = −(δ + iκ/2)ψ + U |ψ |2ψ + $. (3)

The steady-state equation ψ = 0 can be recast as a cubicequation for the mean-field density n = |ψ |2,

n[(δ − Un)2 + κ2/4] = $2. (4)

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FIG. 2. (a) The mean-field phase diagram for a single cavity isdivided into regions that support either one or two dynamically stablesolutions. For this plot and those that follow, all parameter valuesare given in units of δ, and U = 0.1. (b) κ = 0.6: A cut throughthe bistable region of the mean-field phase diagram, showing boththe mean-field (dashed blue curve) and exact (solid orange curve)solution for the density. (c) κ = 0.6 and $ = 1.2: Full countingstatistics of the exact steady-state density matrix, together with thatof both mean-field solutions (i.e., coherent-state distributions, withthe relative normalization used as a fitting parameter). (d) κ = 0.6and $ = 1.2: One trajectory obtained from a quantum-trajectoriessimulation of Eq. (2) for a single cavity, showing switching betweentwo mean-field-like states; the densities associated with the twodynamically stable mean-field solution are shown as dashed lines.

This equation has either one or two solutions that aredynamically stable to small perturbations, leading to themean-field phase diagram shown in Fig. 2(a).

For a single cavity, the full quantum solution of themaster equation can be obtained efficiently in a variety ofways, for example by direct numerical integration of Eq. (2)within a truncated Hilbert space. Steady-state expectationvalues can even be obtained analytically by mapping thesingle-cavity version of Eq. (2) onto phase-space equations inthe complex-P representation [67,75]. Reference [67] providesa comprehensive discussion of the solution, and here wesimply summarize its main features, focusing primarily onthe relationship between the exact and mean-field solutions.While the mean-field equations of motion can support twodynamically stable steady states, the exact steady-state densitymatrix of Eq. (2) is always unique, as are all observablescalculated from it [for example, see Fig. 2(b)]. While there arenever two truly stable steady states, two important signaturesof mean-field bistability do survive in the limit of large cavityoccupancy [76,77]: (1) The full-counting statistics of the exactsolution exhibits a bimodal structure within the parameterregime yielding mean-field bistability, with the probability ofobserving different photon numbers clustering around the twomean-field stable values of the density [Fig. 2(c)]. Outside ofthe bistable region, this bimodality disappears and the exactcounting statistics becomes similar to that corresponding tothe one stable mean-field solution; in this sense, the exactsolution interpolates between the two mean-field steady stateswithin the bistable region. (2) If the system is initialized in

one of the mean-field steady states, it will only explore thephase space in the vicinity of that solution on the natural timescales of the problem (i.e., those associated with energy scalesappearing explicitly in the master equation), and will onlysample the phase space in the vicinity of the other mean-fieldsolution on much longer time scales [Fig. 2(d)]. This slowtime-scale for switching between mean-field-like steady statesis associated with a small gap of the exact quantum Liouvillian,which vanishes inside the bistable region in the limit of largephoton occupancy [48,78,79]. In this limit, which plays a roleanalogous to the “thermodynamic limit” of a spatially extendedsystem [80], mean-field bistability can therefore be identifiedwith the existence of a (zero-dimensional) dissipative phasetransition.

III. FUNCTIONAL-INTEGRAL FORMULATIONAND THE SYSTEM-SIZE EXPANSION

To analyze the steady-state behavior of many coupledcavities, it is convenient to recast the master equation in termsof an equivalent functional integral [81],

Z =∫

Dψ(t)Dϕ(t)W(ψ0,t0)eiS , (5)

with action

S = 2i∑

j

∫ ∞

t0

dt(ϕ∂tψ − ϕ∂t ψ)

−∑

j

∫ ∞

t0

dt(Hw(ψ + ϕ) − Hw(ψ − ϕ))

+iκ∑

j

∫ ∞

t0

dt (2ϕϕ − ϕψ + ϕψ). (6)

The functional integral in Eq. (5) is over all unconstrainedpaths for the variables ψj (t) and ϕj (t), the spatiotemporaldependence of which has been suppressed in Eqs. (5) and (6)[82]. The factor W(ψ0,t0) in Eq. (5) is the Wigner functionat the initial time t0, with ψ0 being shorthand for the set offield variables ψj (0) at the initial time. In this paper we onlyconcern ourselves with steady-state properties, and so we cansafely set t0 = −∞ in the integral limits of Eq. (6) and excludethe dependence on W(ψ0,−∞) from Z . From here forwardthe integral limits of ±∞ in the action are implied and notshown.

The classical Hamiltonian Hw in Eq. (6) is the Weylsymbol of the Hamiltonian in Eq. (1) [83,84]; this shouldbe contrasted with the usual appearance of the Q symbolin the Keldysh functional integral. The appearance of theWeyl symbol is a consequence of the way in which Eqs. (5)and (6) are derived. In particular, we do not apply the usualKeldysh rotation to a coherent-state path integral, but ratherprovide a direct construction of Z in terms of the variablesϕ and ψ (see Appendix A for a detailed derivation). Thisconstructive approach has the benefit of elucidating the deepconnections between the functional-integral treatment of openquantum systems—which is widely employed, and reviewedin Ref. [7]—and more traditional phase-space methods. Thesedifferences notwithstanding, the present approach is perfectlyconsistent with the appearance of the Q function in the usual

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Keldysh approach; the difference is compensated by a slightlydifferent set of rules for computing equal-time observables.

While the master equation can be used to compute arbitraryaverages of time-dependent system operators via the quantum-regression formula [4,85], the functional integral in Eq. (5) issuited to calculating products of operators that are orderedalong a closed-time (Keldysh) contour [86,87]. It is worthnoting, however, that the restriction to calculating a particularclass of time-ordered operators is not required in order toformulate the problem via a functional integral; rather, itenables the functional integral to be formulated along aparticularly simple (Keldysh) time contour. The simplicityafforded by Keldysh time ordering also manifests itself inmore traditional quantum-optics approaches to computingtime-dependent observables: The observables made accessiblevia a functional integral formulated on the Keldysh contourare precisely the same as those computable via the quantum-regression formula without evolving the density matrix back-wards in time.

Writing the so-called classical (ψ) and quantum (ϕ) fieldsas (ψ1,ψ2) ≡ (ψ,ψ) and (ϕ1,ϕ2) ≡ (ϕ,ϕ), and writing cre-ation and annihilation operators as (a1,a2) ≡ (a†,a), Keldysh-ordered correlation functions can be computed as (restoringspatiotemporal indices and defining µ = 1,2)

⟨TK

(· · · aµ

j (t±) · · ·)⟩

=⟨· · ·

(ψ

µj (t) ± ϕ

µj (t)

)· · ·

⟩Z . (7)

On the left-hand side of Eq. (7) the expectation value istaken with respect to the initial density matrix, and theoperators evolve in the Heisenberg picture [88]. The symbolTK time-orders all operators whose time arguments have a “+”superscript, and anti-time-orders those with time argumentsthat have a “−” superscript, placing all of the latter to theleft of all of the former. On the right-hand side of Eq. (7)the expectation value is taken with respect to the functionalintegral Z; i.e., it is computed by inserting the relevant fieldsinto the integrand of Eq. (5) (Z is normalized to unity byconstruction, as it expresses the trace of the density matrix).In the calculations that follow, we exploit a semiclassical limitin which the quantum field is, in a sense to be made precise,parametrically smaller than the classical field; thus we areprimarily interested in correlations of the classical field alone,which can be converted into operator expectation values byinverting Eq. (7),

⟨ψ

µ1j1

(t1) · · · ψµn

jn(tn)

⟩Z = 1

2n

∑

σ=±

⟨TK

(a

µ1j1

(tσ11

)· · · aµn

jn

(tσnn

))⟩.

(8)

While it may appear that such correlation functions can bediscontinuous at coinciding times due to the associated changeof operator ordering on the right-hand side of Eq. (8), itis straightforward to show that this is not actually the case.Instead, when the times approach each other (t1, . . . ,tn → t),the limit of an arbitrary n-point correlation function of theclassical field ψ smoothly approaches the equal-time value

⟨ψ

µ1j1

(t) · · ·ψµn

jn(t)

⟩Z =

⟨(a

µ1j1

(t) · · · aµn

jn(t)

)s

⟩, (9)

where (· · · )s symmetrizes (i.e., Weyl-orders) products ofcreation and annihilation operators [4]. In other words, equal-time correlation functions of the classical field reproduce

the average of Weyl-ordered operator products. For example,the density can be computed from the two-point correlationfunction

⟨ψj (t)ψj (t)⟩Z = ⟨(a†j (t)aj (t))s⟩ = 1

2 ⟨a†j (t)aj (t) + aj (t)a†

j (t)⟩

= ⟨n(t)⟩ + 12 . (10)

The Weyl symbol of H is given by (ignoring additiveconstants, which do not affect correlation functions)

Hw(α) =∑

j

(−αj (J∇2 + µ + U )αj

+ U

2|αj |4 + $(αj + αj )

). (11)

Here ∇2αj ≡ −zαj +∑

⟨k,j⟩ αk is the discrete Laplacian, thelattice coordination number z = 2D in D dimensions, andµ = δ + zJ . Inserting Eq. (11) into Eq. (6) yields the action

S = 2∑

j

∫dt ϕ

(i∂tψ +

(J∇2 + µ + i

κ

2

)ψ

− $ − U |ψ |2ψ)

+ c.c. + 2iκ∑

j

∫dt ϕϕ

+ 2U∑

j

∫dt (ϕϕ + 1)(ψϕ + ψϕ). (12)

Because the functional integral is Gaussian for U = 0, andbecause optical bistability at the mean-field level can occurat arbitrarily small values of U , one might hope that someaspects of the relevant physics can be captured by doingperturbation theory in U . However, this is not the case; whileoptical bistability can indeed occur for small U , it alwaysoccurs when the typical interaction energy seen by a particle,U |ψ |2, is comparable to the other energy scales of the problem.Nevertheless, the action can still be organized around a smallparameter that enables a controlled approximation. To this end,we define rescaled fields and parameters

+ ≡ ϕ√N , , ≡ ψ/

√N , ω ≡ $/

√N , u ≡ UN ,

(13)

in terms of which the action can be rewritten

S = 2∑

j

∫dt +

(i∂t, +

(J∇2 + µ + i

κ

2

),

−ω − u|,|2,)

+ c.c. + 1N

∑

j

∫dt (2iκ++)

+ 1N 2

∑

j

∫dt 2u(++ + N )(,+ + ,+). (14)

The dimensionless parameter N implicitly identifies a one-dimensional family of actions at fixed values of ω, u, κ ,µ, and J , one limit of which (large N ) will be shown toadmit a tractable analysis. Note that the limit N → ∞ atfixed u and ω is not the same thing as the limit U → 0, eventhough the coupling U does become small in this limit. Rather,increasing N amounts to increasing the drive strength $ while

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simultaneously decreasing the coupling U in such a way thatthe typical interaction energy per particle, U |ψ |2, remainsconstant.

To see this, we first evaluate the functional integral in thelimit N → ∞, in which only the first term in the actionsurvives. The functional integral over + can be carried out andyields a functional δ function of the term inside parentheses,thereby enforcing the mean-field equation of motion

i∂t, = −J∇2, − (µ + iκ/2), + ω + u|,|2,. (15)

Note that at this level of approximation, varying N withω and u held fixed leaves the equation of motion for ,invariant. Therefore, as discussed above, the typical interactionenergy seen by each particle, U |ψ |2 = u|,|2, stays fixed. Theonly consequence is that the actual density, |ψ |2 = N |,|2, isenhanced by a factor of N , which therefore sets the overalldensity scale [79].

For N large but finite, the final term on the secondline of Eq. (14) suppresses contributions to the functionalintegral unless + ! N 1/2. The final term can therefore beestimated as +3N−2 + +N−1 ! N−1/2 and can be safelyignored in the large-N limit. At this level of approximation,the functional integral can no longer be solved exactly, but itcan be mapped onto stochastic classical equations by standardtechniques. Decoupling the term that is quadratic in + with aHubbard-Stratonovich transformation,

e−(2κ/N )|+|2 = 2Nκπ

∫d2ζe2i(+ζ++ζ )e−2N |ζ |2/κ ,

the action again becomes linear in +. This time, for fixed ζ ,the functional integral over + enforces the equation of motion

i∂t, = −J∇2, − (µ + iκ/2), + ω + u|,|2, + ζ. (16)

The remaining functional integral over ζ with a Gaussianweight exp(−2N |ζ |2/κ) indicates that we should interpretEq. (16) as a stochastic differential equation, with ζ beingcomplex, Gaussian white noise of variance (restoring spatialand temporal indices)

ζj (t1)ζk(t2) = 1N

κ

2δj,kδ(t1 − t2). (17)

Hence the dynamics of the rescaled classical field ,, to thisorder in 1/N , is governed by a stochastic and dissipativeGross-Pitaevskii equation with parametrically weak noise[89]. Equation (16) can also be derived using phase-spacetechniques; in this context, it would arise as the so-calledtruncated Wigner approximation, an approximation to the(otherwise-exact) multidimensional partial-differential equa-tion governing the time evolution of the Wigner function.However, the functional-integral approach used here makesthe identification and justification of this approximation verytransparent and has the advantage that one can assess theconsequences of the approximations that lead to Eq. (16)within the framework of the renormalization group. In par-ticular, as discussed in Ref. [69], Eq. (16) should reproducethe correct critical exponents for the phase transition exhibitedby the exact steady state of Eq. (2). Unlike in Ref. [69],however, here we have explicitly identified a limit (largeN ) in which Eq. (16) yields asymptotically exact resultsfor microscopic observables and thus can be used to make

quantitative predictions about the behavior of correlationfunctions at the lattice scale (rather than just qualitativepredictions about their long-distance asymptotics). Moreover,the existence of this limit furnishes a more formal justificationfor perturbative renormalization-group analyses.

It is important to realize that, even for a single cavity,Eqs. (15) and (16) must be interpreted with some care in orderto correctly extract steady-state properties in the large-N limit.In particular, there is a sense in which mean-field theory makesincorrect predictions about the steady-state even in the limitof large N . The difficulty can be seen by returning to thefull functional integral; there, it can be shown that the limitsN → ∞ and t → ∞ do not commute when the parametersare tuned to be inside the mean-field bistable region [90].Indeed, if we take the limit N → ∞ first, Eq. (15) is exactat all times, and we are led to conclude (on the same basis asthe analysis in Sec. II) that there are two stable steady states.If we instead take the large-t limit first, we would find [basedon the analysis of 1/N corrections contained in Eq. (16)] thatthere is a unique steady state at any finite value of N , whichis a fluctuation-induced admixture of the two stable steadystates computed by reversing the order of limits. If we nowtake the large-N limit, one of those steady states is generallypreferred over the other, in the sense that it alone determines allsteady-state expectation values. Thus we encounter a sudden(first-order) phase transition between a bright and a dark statewhen traversing through the mean-field bistable region.

IV. MEAN-FIELD THEORY

From the above considerations it is clear that, at leastin the large-N limit, the steady state can be understood bysolving the mean-field equations of motion in the presenceof parametrically weak noise. Thus we expect a detailedunderstanding of the mean-field dynamics in the absence ofnoise to form a useful starting point for understanding thedynamics of Eq. (16). At the level of Eq. (15), and assumingthat only uniform steady states exist, the steady-state phasediagram is identical to that of single-cavity optical bistability,up to the replacement of δ by µ. In terms of the rescaled field ,and a rescaled density N ≡ |,|2 = N−1|ψ |2, Eq. (4) becomes

N ((µ − uN )2 + κ2/4) = ω2. (18)

Straightforward analysis of Eq. (18) shows that upon enteringthe bistable region from outside of it, the additional solutiondoes not in general emerge continuously from the existingone. However, if one enters the bistable region through thecusp located at (see Fig. 3)

{κc,ωc} = µ{(4/3)1/2,(2/3)3/2(µ/u)1/2}, (19)

the two solutions do emerge continuously from a singlesolution, ,c = e−iπ/3√2µ/3u (with critical density Nc =|,c|2 = 2µ/3u). Therefore, we can identify the cusp ofthe bistable region as a mean-field critical point locating acontinuous phase transition from one to two steady states.

It is straightforward to show that one can only enterthe bistable region through the critical point along the line(κ − κc) =

√8u/µ(ω − ωc). It is convenient in what follows

to define new coordinates in the κ-ω parameter space thatnaturally parametrize deviations from the mean-field critical

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FIG. 3. (a) Coordinates used to parametrize passage through themean-field critical point and into the bistable region; r and h controldeviations in a nonorthogonal coordinate system spanned by thedashed arrow. (b) Setting h = 0 and scanning r from positive tonegative causes the order parameter to undergo a cusp bifurcation, atwhich the solution outside the bistable region goes unstable (dashedline) and two new dynamically stable steady states (solid lines)emerge continuously. In both plots u = µ, and energies are givenas dimensionless ratios with µ.

point along and away from this line,

r = 12

(κ − κc), h = 4√3

(ω − ωc) −√

2µ

3u(κ − κc). (20)

Note that this parameter transformation is designed so thatonly r varies as we enter the bistable region through the criticalpoint—the overall normalization of r and h is arbitrary, andchosen to make the formulas that follow simpler. These newvariables can be visualized as parametrizing deviations fromthe mean-field critical point in the nonorthogonal coordinatesystem shown in Fig. 3(a). For h = 0, moving from r > 0 tor < 0 causes the mean-field solution outside of the bistableregion to undergo a cusp bifurcation [Fig. 3(b)]. For r < 0,sweeping h from negative to positive traverses the bistableregion in such a way that the system goes from supportingonly a low-density solution, to having coexisting low-densityand high-density solutions, and then eventually to supportingonly a high-density solution. This behavior is in close analogyto that of an Ising model: If the dark and bright solutionsare identified with the up-down-symmetry-related free-energyminima, then r plays the role of the reduced temperature andh plays the role of a symmetry-breaking (longitudinal) field,causing one to be preferred over the other. The remainder ofSec. IV formalizes this analogy, and in Sec. V we argue that itcontinues to hold even when fluctuations are included.

A. Near-critical dynamics

We are particularly interested in the effects of fluctuationsin the vicinity of the mean-field critical point, which requiresthat we first understand the mean-field response when (a) theparameters are tuned close to the mean-field critical point (bothr and h are small) and (b) the order parameter , is perturbedweakly from its steady-state value. First working directly at thecritical point (r = h = 0, with steady-state solution , = ,c)and assuming that , = ,c + δ, is uniform and close to thecritical value, we expand Eq. (15) to first order in δ, to obtain

∂tδ, = −µ

2[(

√3 + i)δ, + (

√3 − i)δ,]. (21)

The right-hand side (rhs) of Eq. (21) is purely real, and thusonly the real part of δ, decays; the dynamics stops whenthe rhs vanishes, i.e., when arg(δ,) = π/2 − arg(

√3 + i) =

π/3. This observation motivates the following decompositionof the complex-valued δ, into two real components,

δ, = ϱ + eiπ/3σ, (22)

with the expectation that ϱ and σ will relax quickly andslowly, respectively, in the vicinity of the mean-field criticalpoint. Inserting this decomposition into Eq. (15) but nowkeeping all orders in ϱ and σ , we obtain coupled nonlineardifferential equations for ϱ and σ (see Appendix B for adetailed discussion). The fast variable ϱ can be adiabaticallyeliminated by solving ∂tϱ = 0 for ϱ (perturbatively in r andh) and inserting the solution into the equation of motion for σ .In this way, to lowest nontrivial order in r we obtain

∂tσ = J√3∇2σ − rσ − u√

3σ 3 − h

2. (23)

Note that we have also dropped higher-order derivative termsfor σ that arise from the adiabatic elimination of ϱ; thisomission turns out to be justified (to lowest nontrivial orderin r) near the mean-field critical point, where the fields varyslowly in space, even when J is not small compared to theother energy scales in Eq. (23) (the perturbative adiabaticelimination of ϱ is explained in detail in Appendix B).Restoring spatial indices and defining parameters

K = J/√

3, g = u/√

3, (24)

Eq. (23) can be rewritten as

∂tσj = −∂H (σ )∂σj

, (25)

where the effective Hamiltonian H (σ ) is defined as

H (σ ) = 12

∑

j

(K|∇σj |2 + rσ 2

j + 12gσ 4

j + hσj

). (26)

Note that, as anticipated, H (σ ) is precisely the energyfunctional defining the Landau theory of a classical Isingmodel, with σ playing the role of the magnetization. Equation(25) indicates that the dynamics of the slow field in the vicinityof the critical point is purely relaxational, evolving towards theminimum of the effective potential H (σ ).

B. Domain walls

At the level of mean-field theory there are two truly stablehomogeneous solutions within the bistable region. However, itis clear that if we place the system in one of the two mean-fieldsteady states, the inclusion of fluctuations will seed defects ofthe other steady state; whether these defects shrink or grow willdepend on the dynamics of the domain wall separating themfrom the bulk, and will determine which of the two mean-fieldsteady states is preferred over the other. Thus the identificationof a point in the bistable region where the mean-field velocityof a domain wall vanishes gives a first approximation to thelocation of the phase transition when fluctuations are included.

It is difficult to analytically extract domain-wall dynam-ics directly from Eq. (15), so to proceed we make three

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assumptions: (1) the parameters are tuned to be inside thebistable region and close to the mean-field critical point, (2)the domains are smooth, such that a continuum approximationis justified, and (3) if D > 1, the domains are large and thushave vanishing curvature. The first assumption justifies the useof Eq. (25) to calculate the dynamics. The second assumptionrequires that J is large compared to the local energy scalesof the problem, e.g., to the characteristic time scale associatedwith dynamics in the potential part of the Hamiltonian

U (σ ) = 12

(rσ 2 + 1

2gσ 4 + hσ). (27)

Note that near criticality, this only requires that J is largecompared to r and h, and not that J is large compared to theenergy scales ω and γ . The third assumption is made becausethe phase that is favored in the limit of weak fluctuations isthe one in which asymptotically large defects of the oppositephase are unfavored (i.e., tend to shrink).

The dynamics of a flat domain wall is effectively one-dimensional and can be ascertained from a one-dimensionalcontinuum version of Eq. (25),

∂tσ (x,t) = K∂2xσ (x,t) − rσ (x,t) − gσ (x,t)3 − h

2. (28)

When h = 0, the symmetry of Eq. (28) under inversions σ →−σ implies that domain walls must be stationary; both uniformphases have the same effective potential, and relaxationaldynamics cannot prefer one over the other. Therefore, theline h = 0 provides a first approximation to the dividing linebetween parts of the bistable region in which the bright phaseis more stable and parts in which the dark phase is more stable.

For h small but nonzero, the domain-wall velocity can beestimated in the following manner [91]. Making a travelingwave ansatz σ (x,t) = σ (τ ), with τ ≡ x − vt , and denotingderivatives with respect to τ by dots, Eq. (28) becomes

K σ = −vσ + ∂U (σ )∂σ

. (29)

Equation (29) can be interpreted as Newton’s equation for aparticle with position σ and mass K , moving in the invertedpotential −U (σ ) and subject to a linear drag with frictioncoefficient v. Inside the mean-field bistable region (r < 0)there are two stationary solutions of Eq. (29) associated withthe two local maxima of the potential energy −U (σ ) (Fig. 4);these correspond to the two spatially uniform mean-fieldsteady states. To zeroth order in h, these solutions are located at

FIG. 4. (a) Domain-wall dynamics near the critical point. Sincethe dynamics is relaxational, the domain wall moves in such adirection that the lower-energy domain increases in size. (b) Thisdynamics can be mapped onto the motion of a fictitious particle in aninverted potential.

FIG. 5. (a) Numerically determined location of the domain-wallvelocity zeros (red disks), together with the analytical estimate ofthe zero-velocity line, h = 0, valid near the critical point (red line).The size of the disks reflects the largest expected uncertainty in thenumerical determination of these points. Inset: Exploded view of themain plot near the critical point, now in terms of the parameters h andr . The dashed blue line is an improved estimate of the zero-velocityline obtained by extending Eq. (23) to next leading (quadratic) orderin r . (b) Numerically extracted domain-wall velocity (red disks) asa function of h [taken along the black dotted line shown in the insetof (a)], compared with the estimate in Eq. (30) (black dashed line).Note that the black dashed line does not vanish at h = 0. While itsslope is taken from Eq. (30), it has been shifted by an amount thatwe determine by extending Eq. (23) to next-to-leading order in thedeviations from the critical point [i.e., the same correction used toproduce the blue dashed line in the inset of (a)], which clearly agreeswell with the numerically calculated shift of the zero-velocity point.

σ± = ±σ0, with σ0 =√

|r|/g. In addition to the two stationarysolutions, a solution can be found that interpolates from thehigher local maximum to the lower one, which for h > 0is located at σ = +σ0. The friction coefficient v must bedetermined self-consistently such that the particle comes torest at the lower local maximum. Standard analysis of thesolutions of Eq. (29) based on conservation of energy yields,to first order in h (see Appendix C for details),

v ≈ h32

√Kg

2r2. (30)

The analysis above is corroborated by brute-force numericalintegration of Eq. (15). The true zero-velocity line can bedetermined numerically by solving Eq. (15) with a domainwall inserted at t = 0, and agrees with the h = 0 line nearthe critical point [Fig. 5(a)]. Also, as shown in Fig. 5(b),Eq. (30) agrees well with the numerically extracted domain-wall velocity.

V. LANGEVIN EQUATIONS

Mean-field theory suggests that the steady state of themaster equation in Eq. (2) undergoes an Ising-like phasetransition in sufficiently high spatial dimensions. However, inorder to understand the detailed nature of this phase transition,and to determine its lower critical dimension, fluctuations mustbe taken into account. As discussed in Sec. III, for large Nthe dominant fluctuations are captured by working with thestochastic and dissipative Gross-Pitaevskii equation (GPE) in

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Eq. (16), reproduced here for clarity:

i∂t, = −J∇2, − (µ + iκ/2), + ω + u|,|2, + ζ. (31)

At this level of approximation, expectation values of theclassical field are obtained by averaging the solution of Eq. (31)over realizations of the noise ζ , ⟨· · · ⟩Z ≈ ⟨· · · ⟩sGPE. Note that,unlike in mean-field treatments, Eq. (31) gives access to ap-proximate correlation functions of operators at unequal pointsin space and time via the correspondence in Eqs. (8) and (9).

Though we cannot solve Eq. (31) analytically, simplearguments can be made to explain many features of thesteady state quantitatively near the mean-field critical point,and qualitatively even away from it. As before, the near-critical dynamics is simplified by decomposing the field as, = ,c + (ϱ + eiπ/3σ ). Adiabatic elimination of ϱ can againbe performed perturbatively in h and r; the only subtlety isthat fluctuations cause ϱ to undergo a lattice version of theOrnstein-Uhlenbeck process [92], which feeds back into theequation of motion for σ as non-δ-correlated noise. However,it is straightforward to show that near the critical point thecorrelation time of this additional noise is short compared tothe dynamical time scales of σ , and it can be incorporatedas a perturbative renormalization of the δ-correlated noiseacting directly on σ . Details of the calculation are reportedin Appendix B, and here we simply quote the final result,

∂tσj = −∂H (σ )∂σj

+ ξj (t). (32)

Here, H (σ ) is the same energy functional given in Eq. (26),and ξj (t) is (real) Gaussian white noise with variance

ξj (t1)ξk(t2) = κ

3Nδj,kδ(t1 − t2). (33)

Equation (32) is a spatially discretized version of modelA in the Hohenberg-Halperin classification [93], suggestingthat the steady-state phase transition associated with opticalbistability in the driven-dissipative Bose-Hubbard model is, asanticipated, in the universality class of the finite-temperatureclassical Ising model. In particular, steady-state and staticobservables generated by the stochastic dynamics in Eq. (32)can be computed with respect to a Boltzmann weight,

P(σ ) = Z−1e−H (σ )/Teff , Z =∫ ∏

j

dσj e−H (σ )/Teff ,

(34)

with an effective temperature given by

Teff = κ/3N . (35)

The large-N limit was designed to suppress fluctuations in themicroscopic action and so, unsurprisingly, it corresponds to alow-temperature limit of the effective equilibrium descriptionof the phase transition. Returning to the underlying micro-scopic degrees of freedom, it is straightforward to see that thedynamics of this effective theory is imprinted on experimen-tally measurable observables. The connection is particularlysimple near the mean-field critical point. For example, workingto lowest nontrivial order in the deviations of the fields fromtheir mean-field critical values, straightforward algebra yields

the equal-time connected density-density correlation function

Cjk = ⟨nj (t)nk(t)⟩ − ⟨nj (t)⟩⟨nk(t)⟩∝ ⟨σj (t)σk(t)⟩sGPE − ⟨σj (t)⟩sGPE⟨σk(t)⟩sGPE. (36)

The critical properties of the finite-temperature Ising modelshould, therefore, control the critical fluctuations of theintensity of light emitted from a coherently driven array ofexciton-polariton microcavities.

Before considering what happens away from the criticalpoint, we first briefly summarize a qualitative picture of modelA dynamics and its connection to the Ising model. Suppose thatthe system is seeded in a locally random initial configuration:We would like to know what happens to it in steady state.At short times and for r < 0, we expect the system to formdomains of both (locally stable) phases, separated by domainwalls. In the absence of fluctuations (i.e., at Teff = 0) the pre-ferred steady state of the system can be understood by simpledomain-wall dynamics; for h = 0 one phase is preferred overthe other, and the system will eventually order in that phase.If fluctuations are now turned on, domains of the less favoredphase will be seeded, and the consequence of these defectsdepends crucially on the dimensionality. In one dimension,the domain walls enclosing these defects move independentlyof each other when they are sufficiently far apart, undergoinga biased random walk. As a result, when h → 0 and the dy-namics becomes unbiased, defects proliferate and the systemwill be disordered at any finite temperature. In two or morespatial dimensions, defects of the less favored phase will still beseeded by fluctuations, but small defects contract aggressivelyeven as h → 0 due to a surface tension. Therefore, at least atsufficiently small temperature, as h → 0 the system remainsordered in a phase that depends on whether h approaches zerofrom below or above, indicating a first-order phase transition.

Since the above argument relies very little on the dy-namics being relaxational, and primarily on the existence ofdomain walls that—in the absence of fluctuations and forasymptotically large domains—have a directional preferencethat changes as we move through the bistable region, it isreasonable to expect the qualitative picture described aboveto be valid even away from the critical point. Nevertheless,because the dynamics generated by Eq. (16) does not induce anequilibrium steady-state distribution far away from the criticalpoint, it is important to verify this picture numerically.

To this end, we carry out a brute-force numerical integrationof Eq. (31) in both one and two dimensions using a fixed-time-step first-order Euler-Mayurama method. After a burn-intime, the equations are integrated until statistical error bars(1σ ) of fractional size 0.01 are achieved for the density.Temporal autocorrelations on time scales of 1/10 the totalintegration time are also required to fall below a similarthreshold to ensure that the averaging time is long comparedto all dynamical time scales, which can become anomalouslylarge near the crossover or phase transition. Exemplary resultsof these numerics in one dimension are shown in Fig. 6 andreflect the spatiotemporal dynamics of output light intensitythat would be observed if the model were realized in an arrayof exciton-polariton microcavities. As expected, by sweepingvertically through the mean-field bistable region, we changefrom a dominantly dark steady state with small domains of

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FIG. 6. Real-time dynamics (after burn in) in a 1D systemwith 128 sites and periodic boundary conditions, showing domainproliferation in the vicinity of the crossover in one dimension. In allthree plots, (J,U,κ) ≈ (0.1µ,0.2µ,0.3µ). The left-hand panel is juston the dark side of the crossover, the middle panel is roughly in themiddle of the crossover, and the right-hand panel is just on the brightside of the crossover. The color indicates the density.

the bright phase to a predominantly bright steady state withsmall domains of the dark phase. Because the domain wallsare unbound, this change manifests as a smooth crossoverrather than a true phase transition, as confirmed in the 1Dphase diagram shown in Fig. 7(a) [in particular, see the crosssection plotted in Fig. 7(c)]. Near the mean-field critical point,the characteristic domain size at the crossover point (see, forexample, the central panel of Fig. 6) reflects the effectivetemperature of the model; as κ shrinks the domains grow insize, and the crossover becomes sharper. However, the extent towhich this behavior persists as κ → 0, and whether this limitis strictly analogous to the T → 0 limit of the Ising model,is difficult to say. A careful numerical analysis of density-density correlation functions in one dimension reveals thatthey always decay exponentially at the crossover point, with acorrelation length that grows monotonically with decreasing κ .However, as κ decreases, eventually the domain size becomesso large—and the dynamics of domain-wall diffusion becomesso slow—that we are unable to obtain statistically convergedresults. This computational limitation imposes the lower limiton κ in the 1D phase diagram reported in Fig. 7(a).

In two dimensions [Figs. 7(b) and 7(d)], the dynamics isqualitatively different. In sweeping from small to large drivesat sufficiently small κ , one encounters a clear first-order phasetransition between the bright and dark phases, consistent withthe expected equilibrium physics of the 2D Ising model. Thesize of the discontinuity increases with decreasing κ (andthus with decreasing effective temperature). The initial stateis a random admixture of the two mean-field steady states,which plays the role of an infinite-temperature state. Thusfor κ below the critical point and $ chosen close to thefirst-order phase transition, the initial dynamics can be viewed(in the language of equilibrium physics) as a quench froman infinite-temperature phase to a final temperature belowthe ordering temperature. The short-time dynamics thereforeshows the expected coarsening of small domains. Eventuallythe more favored phase wins out unless $ is tuned precisely

FIG. 7. Phase diagrams obtained by numerically solving Eq. (16)on (a) a 1D chain with 128 sites and (b) a 2D square lattice with32 × 32 sites (in both cases periodic boundary conditions wereused). For both plots, the parameters used are (J,U ) ≈ (0.1µ,0.2µ).The solid white line locates the (near-critical) condition for avanishing domain-wall velocity, h = 0, while the white circlesindicate numerically obtained velocity zeros. The dashed white linesindicate the parameter regime used for plots (c) and (d). Note that thephase diagrams are cut off at small κ , or equivalently low effectivetemperature, because statistically converged numerical solutions ofEq. (16) require prohibitively long integration times as fluctuationsbecome weaker. (c) and (d) Cuts through the phase diagrams indicatedby dashed white lines in (a) and (b). In one dimension (c) themean-field phase transition is smoothed out into a crossover, whilein two dimensions (d) bistability leads to a true first-order phasetransition. The error bars in (c) and (d) are smaller than the size ofthe plot markers.

to the phase transition, but this claim takes progressivelymore averaging to establish reliably as one moves closer tothe phase-transition line. Once again, decreasing the effectivetemperature by decreasing κ leads to a slowing down of thedynamics, making it difficult to obtain statistically convergedresults near the first-order phase transition and imposing alower limit on κ in the 2D phase diagram reported in Fig. 7(b).

VI. DISCUSSION

By bringing together a number of ideas from both quantumoptics and condensed-matter physics, we have identified alimit of the driven-dissipative Bose-Hubbard model in whichthe dominant fluctuations are captured by nonequilibriumLangevin equations, enabling a quantitatively accurate andcomputationally efficient determination of steady-state prop-erties. Near the critical point, these fluctuations are thermaland lead to an effective equilibrium description. However,we emphasize that the Langevin description of the problemshould be asymptotically exact in the limit of N → ∞ evenaway from the critical point, where an equilibrium descriptionis not valid. Numerically, we find that in two dimensions thefirst-order phase transition expected from the mapping onto an

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Ising model remains intact far from the critical point, wherethis mapping is not strictly valid. In addition, the absenceof a phase transition in one dimension is the result of thesame domain-wall phenomenology that prevents ordering ofthe 1D Ising model at finite temperatures. In this way, thenumerical results reinforce and extend the assertion that thesteady-state behavior of the driven-dissipative Bose-Hubbardmodel possesses an emergent description in terms of theequilibrium physics of a finite-temperature classical Isingmodel. These conclusions have direct consequences for a rangeof experiments in which particle loss is countered by coherentdriving, for example a coherently driven exciton-polaritonfluid in an array of semiconductor microcavities. Here, bytuning the laser driving strength through the mean-fieldbistable regime, one should be able to observe domain growth,hysteresis, critical fluctuations, and other generic features ofthe Ising model in a longitudinal field, all by simple correlationmeasurements on the output intensities of the cavities.

We caution that any claims about the universality class ofthe phase transition require more than just a microscopicallyaccurate treatment of fluctuations—it is also important toidentify the relevance of any ignored fluctuations in thesense of the renormalization group, even if they are para-metrically small. In other words, there is no guarantee thatsmall (1/N ) quantitative errors at the scale of the latticespacing will not qualitatively affect the nature of the phasetransition. Renormalization-group arguments (i.e., canonicalpower counting) supporting the Ising universality class as thecorrect critical theory at small but finite N can be inferredfrom the results of Ref. [69]. Moreover, we note that becauseEq. (32) provides an increasingly accurate approximation tothe microscopic dynamics for increasing N , Ising-like criticalbehavior should manifest itself at least as an intermediatelength-scale crossover phenomenon, regardless of the trueuniversality class of the phase transition. We also emphasizethat the precise nature of the first-order phase transition faraway from the critical point is less clear, in particular its fateas κ → 0. It would be worthwhile to extend the numericalapproach taken in Sec. V to confirm the universal aspects ofboth the critical point and the first-order phase transition.

It would also be worthwhile to compare some of theresults in one dimension with numerically exact calculationsbased on the density-matrix renormalization group [94] inorder to better understand the importance of higher-order(in 1/N ) corrections that are not captured by the Langevindescription. In particular, at large U (small N ), mean-fieldarguments suggest that the inclusion of fluctuations ignored toleading order in 1/N may lead to richer steady-state behaviors[47,58], including phases that spontaneously break discretespatial-translation symmetry [47,48]. In two dimensions, theformalism described here could be used to compute otherdynamical aspects of the system near the first-order phasetransition; for example, it should be possible to calculate thelifetime of the metastable phase via an instanton approach.

ACKNOWLEDGMENTS

We thank Cristiano Ciuti, Sebastian Diehl, HowardCarmichael, Sarang Gopalakrishnan, Victor Gurarie, AnaMaria Rey, Murray Holland, Anzi Hu, Michael Fleischhauer,

and Chih-Wei Lai for helpful discussions. M.F.M., J.T.Y., andA.V.G. acknowledge support by ARL CDQI, ARO MURI,NSF QIS, ARO, NSF PFC at JQI, and AFOSR. R.M.W.acknowledges partial support from the National Science Foun-dation under Grant No. PHYS-1516421. M.H. acknowledgessupport by AFOSR-MURI, ONR, and the Sloan Foundation.

APPENDIX A: FUNCTIONAL-INTEGRAL FORMALISM

The functional integral presented in Sec. III is closelyrelated to the usual Keldysh functional-integral formalism,for which there are many good references (see, for example,Ref. [87]). However, there is a subtle difference betweenthe formalism used here and that typically employed inthe condensed-matter community, and it therefore seemsworthwhile to provide an explicit derivation of Eqs. (5)–(9).For simplicity, we treat only the single-cavity case, but thegeneralization to many cavities that yields Eqs. (5)–(9) followsimmediately.

In any functional-integral formulation of quantum me-chanics, operators must be traded in for classical variables.The usual way to do this, as is the case in the standardapproach to the Keldysh function integral, is to repeatedlyinsert coherent-state resolutions of identity during the timeevolution. Operators get sandwiched between coherent states,and if they are normal ordered they turn into functions ofphase-space variables. In the language of quantum optics,operators are exchanged for their Q symbols. However, thereare many different ways to associate operators with functionsover phase space, and thus many ways to formulate a functionalintegral. In the following derivation, we replace operatorswith classical variables by working in the Weyl representation(see, for example, Ref. [81]). In our opinion, even thoughthis strategy entails some additional overhead in phase-spaceformalism, it is both more direct and conceptually simplerthan the usual approach to the Keldysh functional integral. Inparticular, the canonical approach described in Refs. [86,87]relies on the construction of a formal continuous-time notationthat—together with simple rules for computing equal-time cor-relation functions—correctly reproduces the continuous-timelimit of the Green’s functions of a noninteracting Bose field.Interactions are then included in a self-consistent fashion byensuring that the rules for Gaussian integration produce correctresults for the interacting theory at all orders of perturbationtheory. In the approach taken here, the functional integral isderived constructively in such a way that an unambiguouscontinuous-time notation emerges naturally from a properlydefined (i.e., discretized) functional integral.

1. Functional representation of the Wigner function

The Weyl symbol of an arbitrary operator A, denotedAw(ψ), can be defined via the relation

Aw(ψ) = Tr[δw(ψ − a)A]. (A1)

Here, the Weyl-ordered (and operator-valued) delta functionis defined by

δw(ψ − a) = 1π2

∫d2ϕ exp[ϕ(ψ − a) − ϕ(ψ − a†)]. (A2)

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When convenient, the correspondence between operators andtheir Weyl symbols is indicated below with the notationA ↔ Aw(ψ). Given the special role played by the densityoperator ρ, it is traditional to use a special notation for its Weylsymbol, W(ψ,t), which is also called the Wigner function; theexplicit time dependence is included because we will workin the Schrödinger picture, where the density matrix (andtherefore the Wigner function) evolves in time.

The Weyl representation is intimately related to symmetri-cally ordered operator products; if an arbitrary operator A isexpanded in terms of symmetrically ordered operator products,

A =∑

p,q

Apq[ap(a†)q]s, (A3)

then the coefficients in this expansion determine the Weylsymbol Aw(ψ) in a particularly natural way:

Aw(ψ) =∑

p,q

Apqψpψq . (A4)

Given the Wigner function at an initial time t0, we wouldlike to understand how it has changed a short time δt later dueto the evolution of the density matrix by the master equation.During this time, the density matrix evolves according toρ(t0 + δt) = Vδt (ρ(t0)), where the infinitesimal time-evolutionsuperoperator Vδt satisfies

Vδt (⋆) = 1 − iδt[H ,⋆] + δtκ

2

∑

j

(2aj ⋆ a†j

− ⋆ a†j aj − a

†j aj⋆) + O(δt2). (A5)

This transformation induces a corresponding evolution of theWigner function, which for now we write formally as

W(ψ1,t0 + δt) =∫

d2ψ0 V(ψ1,ψ0)W(ψ0,t0), (A6)

thereby implicitly defining the infinitesimal phase-space prop-agator for the Wigner function, V .

From the structure of Eqs. (A5) and (A6), it is clear thatfinding the explicit form of V requires us to compute theWeyl symbol of products of ρ with creation and annihilationoperators. To this end we define an operator-valued generatingfunction

G = eηa+ηa†ρ(t0), (A7)

which can be differentiated to produce symmetrically orderedoperator products

∂pη ∂

qη G

∣∣η=0 = [ap(a†)q]sρ(t0). (A8)

Using Eqs. (A3) and (A8), we can expand the product of anarbitrary operator with the density matrix as

Aρ(t0) =∑

p,q

Apq∂pη ∂

qη G

∣∣η=0, (A9)

giving us a prescription to compute the Weyl symbol of Aρ(t0)from the Weyl symbol of G,

Aρ(t0) ↔∑

p,q

Apq∂pη ∂

qηGw(ψ1)

∣∣η=0. (A10)

Making use of the standard operator phase-space correspon-dences [4],

aρ ↔(ψ + 1

2∂ψ

)W(ψ), a†ρ ↔

(ψ − 1

2∂ψ

)W(ψ),

ρa ↔(ψ − 1

2∂ψ

)W(ψ), ρa† ↔

(ψ + 1

2∂ψ

)W(ψ),

the commutation relations [∂ψ ,ψ] = 1, [∂ψ ,ψ] = 0, and theBaker-Campbell-Hausdorff formula, the Weyl symbol of Gcan be written

Gw(ψ1) = e12 η∂ψ1

− 12 η∂ψ1 eηψ1+ηψ1W(ψ1,t0). (A11)

Inserting a standard representation of the δ function andintegrating by parts, we obtain

Gw(ψ1) = 4π2

∫d2ϕ0d

2ψ0e2ϕ0(ψ1−ψ0)−2ϕ0(ψ1−ψ0)

× eη(ϕ0+ψ0)+η(ϕ0+ψ0)W(ψ0,t0). (A12)

Note that the choice of a generating function that producedsymmetrically ordered operator products also led to all of thederivatives appearing on the left in Eq. (A11), which enabledthe integration by parts to proceed in a particularly simplemanner to obtain Eq. (A12). Inserting Eq. (A12) into Eq. (A10)and then using Eq. (A4), we obtain

Aρ ↔ 4π2

∫d2ϕ0d

2ψ0e2ϕ0(ψ1−ψ0)−2ϕ0(ψ1−ψ0)

× Aw(ψ0 + ϕ0)W(ψ0,t0). (A13)

Given Eq. (A13), we can now deduce Eq. (A6) fromEq. (A5). To first order in δt we find

W(ψ1,t0 + δt) = 4π2

∫d2ψ0d

2ϕ0eiδtL(ψ1,ϕ1;ψ0,ϕ0)W(ψ0,t0),

(A14)

where

L(ψ1,ϕ1; ψ0,ϕ0) = 2iϕ0(ψ1 − ψ0)/δt − 2iϕ0(ψ1 − ψ0)/δt

−Hw(ψ0 + ϕ0) + Hw(ψ0 − ϕ0)

+iκ(2ϕ0ϕ0 − ϕ0ψ0 + ϕ0ψ0). (A15)

The Wigner function at a general time t can be obtained fromthe Wigner function at time t0 by iteration of Eq. (A14).Breaking the interval [t0,t] into N segments of size δt =(t − t0)/N , we obtain

W(ψN,t) =∫ N−1∏

j=0

(4π2

d2ψj d2ϕj

)eiSW(ψ0,t0). (A16)

Here we have defined the discretized action

S =N−1∑

j=0

δtL(ψj+1,ϕj+1; ψj ,ϕj ). (A17)

Defining functional-integration measures that include thefields ψN,ϕN at the final time,

Dψ =N∏

j=0

d2ψj , Dϕ =N∏

j=0

4π2

d2ϕj , (A18)

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FIG. 8. Time ordering of operators in Eq. (A20).

the trace of the Wigner function at time t can now be written

Z ≡∫

DψDϕeiSW(ψ0,t0) = 1. (A19)

The continuous-time limit (N → ∞,δt → 0) of Eqs. (A17)–(A19), generalized to many coherently coupled bosonicmodes, yields Eqs. (5) and (6) of the main text.

2. Expectation values

The functional-integral representation of the Wigner func-tion lends itself naturally to calculating correlation functionsthat are time ordered along the Keldysh contour (Fig. 8), e.g.,

C = ⟨TK (B1(t−1 ) · · · Bn(t−n )A1(t+1 ) · · · Am(t+m ))⟩. (A20)

Here, C is written in the Heisenberg picture, and TK ordersoperators at times with a “+” superscript such that the timesincrease from right to left, and orders operators at times with a“−” superscript such that the times increase from left to right.Note that the Heisenberg picture is to be interpreted beforethe adiabatic elimination of the reservoir, which we assumeproceeds within the Born-Markov approximation. Assumingwithout loss of generality that t±j > t±j−1, insisting that theoperators be inserted at the discretized time slices chosen above(t±j = δt × r±

j , with r±j an integer between zero and N ), and

making the notational change A(δtr±j ) → A(r±

j ), C can bewritten

C = Tr(Am(r+m ) · · · A1(r+

1 ) ρ(t0) B1(r−1 ) · · · Bn(r−

n )). (A21)

For our purposes we need to rewrite this correlation functionin the Schrödinger picture, which can be accomplished withthe help of the quantum-regression formula. For example, ift+1 < t−1 , we have [4]

C = Tr(· · · Uδt(r−

1 −r+1 )

(A1Uδtr+

1(ρ(t0))

)B1 · · ·

), (A22)

where Uδtj = Vjδt is the time-evolution superoperator. The

choice of Keldysh ordering is necessary and sufficient toguarantee that, in the quantum-regression formula, it willnever be necessary to evolve backwards in time. Using thefunctional-integral expression for the Wigner function at timet , together with the phase-space correspondences for operatorsA and B, we find

C =∫

DψDϕ e−iSW(ψ0,t0) (A23)

×(· · ·A1

w

(ψr+

1+ ϕr+

1

)B1

w

(ψr−

1− ϕr−

1

)· · ·

)(A24)

≡⟨· · ·A1

w

(ψr+

1+ ϕr+

1

)B1

w

(ψr−

1− ϕr−

1

)· · ·

⟩Z . (A25)

Such correlation functions can be conveniently computed withrespect to a generating functional by adding source terms to

action,

C =(

· · ·A1w

(∂

δt∂Jr+1

+ ∂

δt∂Kr+1

)

× B1w

(∂

δt∂Jr−1

− ∂

δt∂Kr−1

)· · ·

)Z( J,K )

∣∣∣∣J,K=0

.

Here

Z( J,K ) =∫

DψDϕ eiS( J,K )W(ψ0,t0), (A26)

and

S( J,K ) = S +N∑

j=0

δt( J j · ψj + K j · ϕj ), (A27)

with

J j =⎧⎪⎪⎩Jj

Jj

⎫⎪⎪⎭, K j =

⎧⎪⎪⎩Kj

Kj

⎫⎪⎪⎭, ψj =

⎧⎪⎪⎩ψj

ψj

⎫⎪⎪⎭, ϕj =

⎧⎪⎪⎩ϕj

ϕj

⎫⎪⎪⎭.

(A28)

Restricting to the special case when the operators A and B arecreation and annihilation operators, defining

1δt

∂

∂Jj

= δ

δJ (tj ),

1δt

∂

∂Kj

= δ

δK(tj ), (A29)

and taking the continuum limit δt → 0, we recover theexpressions for correlation functions in Sec. III of the maintext.

The correspondence given above can also be reversed insuch a way to rewrite expectation values of the fields ψ andϕ in terms of operator averages. To this end, we rearrange theoperator phase-space correspondences as

{a,ρ} ↔ 2ψW, {a†,ρ} ↔ 2ψW,

[a,ρ] ↔ ∂ψW, [a†,ρ] ↔ −∂ψW.

The substitutions ∂ψ ↔ −2ϕ and ∂ψ ↔ 2ϕ are valid under thefunctional-integration sign, so long as the derivative sits to theleft of any other instances of the field ψ evaluated at the sametime (not including those occurring in the action). Thus we areled to the identifications

ψW ↔ 12 {a,ρ}, ϕW ↔ 1

2 [a,ρ], (A30)

with the understanding that when products of the fields ψ andϕ at the same time arise, we should take the commutators afterthe anticommutators in the corresponding operator expectationvalues. This identification leads very directly to correlationfunctions of the classical field. For example, assuming r3 >r2 > r1 we have

⟨ψr1ψr2ψr3

⟩Z = Tr({a†(r3),{a(r2),{a†(r1),ρ(t0)}}}). (A31)

Note that if r1 = r2 = r3 ≡ r , this correlation function simpli-fies to

⟨ψrψr ψr⟩Z = Tr([a†(r)a(r)a†(r)]s). (A32)

Since we just have a few fields in the correlation function, thisresult can be worked out by direct comparison of the right-handsides of Eqs. (A31) and (A32) (using the commutation relation

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M. FOSS-FEIG et al. PHYSICAL REVIEW A 95, 043826 (2017)

[a,a†] = 1). More generally, it also follows by using the phase-space correspondence [a†aa†]s ↔ (ψ + ϕ)2(ψ + ϕ). Goingback to operator expectation values by using Eq. (A30),and remembering the rule that commutators are taken afteranticommutators when the fields ϕ and ψ are evaluated atthe same times, terms with one or more power of ϕ resultin an outer commutator that gets killed by the trace, so onlythe ψ2ψ term survives. The generalization of Eq. (A32) toarbitrary equal-time correlation functions of the classical fieldψ leads, in the continuum-time limit and for more than onesite, to Eq. (9) of the main text.

APPENDIX B: ADIABATIC ELIMINATIONOF THE MASSIVE FIELD

Here we describe the perturbative adiabatic elimination ofthe field ϱ near the mean-field critical point. For simplicity,we first treat the case with no fluctuations (N → ∞), andafterwards we consider the effect of weak fluctuations.

1. Mean-field theory

Substitution of Eq. (22) into Eq. (15) yields coupledequations for ϱ and σ that are fully equivalent to the mean-fielddynamics of ,. To simplify the following expressions weconvert all energy and time scales into dimensionless ratioswith the chemical potential µ, and in a slight abuse of notationwe do not change any of the associated symbols—the µdependence can be unambiguously restored by insisting ondimensional consistency. After some algebra, we find

ϱ = − 2√3

(ϱ − ϱ0) − rϱ − J∇2

√3

(ϱ + 2σ )

+ u√3

(2σ 3 + 3σϱ2 + 3ϱσ 2 + ϱ3 − σ 2√

6/u), (B1)

σ = −h

2− rσ + J∇2

√3

(σ + 2ϱ)

− u√3

(ϱ2√

6/u + 2ϱ3 + 3σϱ2 + 3ϱσ 2 + σ 3), (B2)

where

ϱ0 = h√

38

− r√8u

. (B3)

Several simplifications can now be made. First, because weare interested in dynamics near the critical point and afterthe field has nearly relaxed, h, r , ϱ, and σ can all be treatedas small parameters. Though we do not know a priori howsmall the fields ϱ and σ are relative to the parameters r andh, it is perfectly consistent to keep, at any particular orderin one of the parameters, only the lowest nontrivial orderin any other parameter. Moreover, while we do not knowhow small spatial derivatives of the field are, we do expectthem to be small compared to the fields themselves, whichshould vary slowly in space near the critical point, and thuswe formally treat J∇2 as an additional small parameter (onecan check that this assumption is self-consistent at the end ofthe calculation, where it is seen that J∇2 ∼ r). Following this

logic, and introducing the rescaled parameters K = J/√

3 andg = u/

√3 used in the main text, we arrive at the simplified

equations

ϱ = − 2√3

(ϱ − ϱ0) − 2K∇2σ + O(σ 2), (B4)

σ = −h

2− rσ + K∇2(σ + 2ϱ) − gσ 3 + O(ϱ2) + O(ϱσ 2).

(B5)

Terms that are kept with the O notation are there to remind usthat we do not know for sure whether they are parametricallysmall compared to other terms that are kept—they turn out tobe unimportant for reasons explained below, which is why wedo not keep track of the exact coefficients.

The justification for adiabatically eliminating ϱ near thecritical point is now clear: As r → 0, the term proportionalto ϱ in Eq. (B4) stays finite, indicating that ϱ relaxes to zeroexponentially in time even at the critical point (once µ isrestored, we see that it decays on a time scale ∼1/µ). On theother hand, the term linear in σ in Eq. (B5) vanishes as r → 0,indicating a divergent time scale for relaxation of σ (whichrelaxes algebraically precisely at the critical point, r = 0). Toadiabatically eliminate ϱ we set the rhs of Eq. (B4) to zero,obtaining

ϱ = ϱ0 −√

3K∇2σ + O(σ 2), (B6)

and then substitute this result into Eq. (B5). Many derivativeterms are generated, but working to lowest order in K∇2 wefind

σ = K∇2σ − rσ − gσ 3 − h

2+ O(r2) + O(rσ 2). (B7)

Here we have implicitly assumed that |h| < |r| to writeϱ0 = O(r), which poses no important restriction on whatfollows. If we ignore the final two terms and solve Eq. (B7)at h = 0 and r < 0 (inside the bistable region), we find twouniform solutions at σ = ±

√|r|/g, which sets the scale of

σ in the relevant near-critical dynamics, σ ∼√

r . From thisscaling, it is easily seen that the final two terms in Eq. (B7) areparametrically smaller than the others; thus we were justifiedin dropping them, which results in Eq. (23) in the main text.

2. Fluctuations

Substitution of Eq. (22) into Eq. (16) yields coupledstochastic equations for ϱ and σ that are fully equivalent to thenonequilibrium Langevin equation for ,. In the limit of weaknoise, an expansion in ϱ and σ is still justified. Moreover, sincearbitrarily weak noise will induce arbitrarily small excursionsaway from the mean-field stationary state, much of the analysisthat led to an effective relaxational description of σ (that isapproximately decoupled from ϱ) should remain valid, as itassumed nothing more than being close to both the criticalpoint and the steady state. Working near the mean-field criticalpoint, the same assumptions that lead from Eqs. (B1) and (B2)to Eqs. (B4) and (B5) remain justified and yield (for now

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keeping all gradient terms)

ϱ = − 2√3

(ϱ − ϱ0) − K∇2(ϱ + 2σ ) + O(σ 2) + η(τ ), (B8)

σ = − h

2− rσ + K∇2(σ + 2ϱ) − gσ 3 + O(ϱ2)

+ O(ϱσ 2) + ξ (τ ). (B9)

In order to avoid confusion regarding the noise variances, herewe have chosen a new symbol for the dimensionless time,tµ ≡ τ (so σ = dσ/dτ , etc.), even though we continue to usethe same symbols for the (now dimensionless) energies r , h,K , and g. The real noises η and ξ are defined as

η(τ ) = ζI(t)µ

− ζR(t)

µ√

3, ξ (τ ) = 2ζI(t)

µ√

3, (B10)

where ζR and ζI are the real and imaginary components,respectively, of the complex Gaussian white noise ζ (t) inEq. (16). Thus from the variances of ζ (t) we have

ηj (τ1)ηk(τ2) = ξj (τ1)ξk(τ2) = κ

3Nδj,kδ(τ1 − τ2), (B11)

where one factor of 1/µ has been absorbed into the nowdimensionless κ and one is used to change variables fromt to τ in the δ function.

We would like to eliminate the terms in Eq. (B9) thatinvolve ϱ, but, strictly speaking, the adiabatic eliminationof ϱ by setting ϱ = 0 in Eq. (B8) is no longer justified.Nevertheless, for weak noise and near the critical point, therewill be a separation of time scales, length scales, and typicalsizes of the fluctuations of σ and ϱ. Because ϱ remainsmassive at the mean-field critical point while σ does not,we expect that in the presence of noise the scale of typicalfluctuations for ϱ will be small compared to the scale oftypical fluctuations for σ . Likewise, σ will relax more slowlythan ϱ near the critical point, and will exhibit fluctuations ona longer length scale, i.e., in the presence of fluctuations itwill have a longer autocorrelation time than ϱ, and will beroughly spatially homogeneous over the length scale on whichϱ is correlated. On the grounds of the latter statement, it isjustified to solve Eq. (B8) at fixed σ and within a local-densityapproximation (i.e., we assume that ϱ is relaxing in a locallyhomogeneous environment set by the slowly varying valueof σ ), in which case ϱ simply undergoes a lattice version ofOrnstein-Uhlenbeck relaxation to the mean value

ϱ = ϱ0 −√

3K∇2σ + O(σ 2). (B12)

Straightforward analysis reveals that fluctuations of ϱ aroundits mean value, ϑ(τ ) = ϱ(τ ) − ϱ(t), obey

ϑj (τ1)ϑk(τ2) ∼ κ

Ne−|j−k|/ℓϱe−|τ2−τ1|/τϱ , (B13)

where ℓϱ ∼√

K and τϱ ∼ 1 are the correlation length andcorrelation time of the massive field ϱ, respectively. Insertingthe solution ϱ = ϱ + ϑ into Eq. (B9), and keeping for now all

terms that involve the fluctuations ϑ , we obtain

σ = −h

2− rσ + K∇2σ − gσ 3 + ξ (τ )

+ 2K∇2ϱ+2K∇2ϑ+O(ϱ2)+O(ϱ)ϑ+O(ϑ2). (B14)

The first and third terms on the second line, 2K∇2ϱ andO(ϱ2), contain terms that are either higher order in σ , in r ,or in gradients than other terms on the first line, and thuscan be ignored. The remaining terms depend on ϑ ; becausethe dynamics of σ is slow and dominated by long-wavelengthfluctuations, we can approximate ϑ as spatially uncorrelatedwhite noise with variance

ϑj (τ1)ϑk(τ2) ∼ κ

Nδj,kδ(τ1 − τ2). (B15)

In light of this approximation, the term K∇2ϑ acts as anadditional source of additive white noise—it can in principlebe taken into account as a (K-dependent) renormalization ofthe noise ξ (τ ) that is already present, but if we work to lowestorder in K this renormalization can be ignored. The termO(ρ)ϑ acts as a source of multiplicative noise; since ϱ is itselfa small parameter, this noise can also be ignored in comparisonto the already present white noise ξ (τ ). The term O(ϑ2) shouldbe interpreted by writing ϑ(τ )2 = f (τ ) + χ (τ ), where f is theaverage of ϑ2 and χ is its fluctuations. The average f ∼ κ/Ncan be ignored for weak noise, while the fluctuations χ obey(using the fact that ϑ is a Gaussian variable)

χj (τ1)χk(τ2) ∼ (κ/N )2e−2|j−k|/ℓϱe−2|τ2−τ1|/τϱ . (B16)

As with ϑ , χ can be interpreted as spatially uncorrelatedGaussian white noise as far as the slow dynamics of σ isconcerned. Since it has a variance that is parametrically smallerthan that of ξ , it can once again be ignored. With all of theterms on the second line of Eq. (B14) dropped, we recoverEq. (32) of the main text.

APPENDIX C: DOMAIN-WALL VELOCITY

As described in the main text, the domain-wall velocitycan be determined by solving for the dynamics of a fictitiousparticle obeying the equation of motion

K σ = −vσ + ∂U (σ )∂σ

, (C1)

where

U (σ ) = 12

(rσ 2 + 1

2gσ 4 + hσ). (C2)

Here the domain-wall velocity v plays the role of a velocity-dependent friction coefficient. We seek solutions of Eq. (C1)for which the particle starts (with σ = 0) at the higher localmaximum of −U (σ ) and comes to rest at the lower one. Whenh is small the two local maxima have similar energies, andthe friction coefficient v must also be small for the particleto reach the top of the lower potential maximum. Thus thetrajectory is similar to that in the case of h = 0, for which(from conservation of energy) K σ 2/2 = U (σ ) − U (σ0), suchthat

σ ≈√

2[U (σ ) − U (σ0)]K

. (C3)

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The work done along this trajectory by the friction must beequal to the change in potential energy,

−v

∫ σ+

σ−

σ dσ = U (σ+) − U (σ−) ≈ hσ0. (C4)

Inserting Eq. (C3) into Eq. (C4), taking the integral, and usingU (σ0) = r2/4g, we obtain the domain-wall velocity given in

the main text,

v ≈ h32

√Kg

2r2. (C5)

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